where \(z= c_{s}k/(aH)\) , \(\kappa =3(H\alpha -\dot{\varphi })+k^{2}\chi /a^{2}\)  and \(\chi =a(\beta +a\dot{\gamma })\) . Moreover, in Eq. (REF )
\(\xi _{q,\Upsilon }\equiv \xi _{q,\Upsilon }({\bf k},t)\)  are stochastic
Gaussian sources related to quantum and thermal fluctuations with
appropriate amplitudes chosen such as to match the analytical
derivation for the scalar of curvature power spectrum in
WI [1]}. This leads in particular that
\(\xi _{\Upsilon }\)  and \(\xi _{q}\)  both have zero mean and they satisfy
two-point correlation functions given, respectively, by
\(\!\!\!\!\!\!\!\!\!\!\langle \xi _\Upsilon ({\bf k},t)\xi _\Upsilon ({\bf k}^{\prime },t^{\prime })\rangle &=& \frac{2 \Upsilon T}{a^3} \delta (t-t^{\prime }) (2 \pi )^3\delta ({\bf k} + {\bf k}^{\prime }),\\ \!\!\!\!\!\!\!\!\!\!\langle \xi _q({\bf k},t)\xi _q({\bf k}^{\prime },t^{\prime })\rangle &=& \frac{H^2(9+12\pi Q)^{1/2}(1+2n_*)}{\pi a^3}\nonumber \\ &\times & \delta (t-t^{\prime }) (2 \pi )^3 \delta ({\bf k} + {\bf k}^{\prime }),\) 
